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Lusin’s condition (N) and mappings with nonnegative Jacobians. (English) Zbl 0807.46032

A continuous mapping \(f: G\to\mathbb{R}^ n\) (\(G\) a domain in \(\mathbb{R}^ n)\) is said to satisfy Lusins’ condition (N) if \(f(A)\) has zero Lebesgue measure whenever \(A\subseteq G\) has zero Lebesgue measure. The author investigates Lusins’ condition for continuous functions in \(W^{1,n} (G,\mathbb{R}^ n)\), noting that in \(W^{1,p} (G,\mathbb{R}^ n)\) with \(p>n\) this condition holds while for \(p<n\) it generally fails. The results are in close connection with those of Yu. G. Reshetnyak. Among other results, the author proves that:
For \(f\in W^{1,n} (G,\mathbb{R}^ n)\) a continuous function with \(Jf\geq 0\) a.e., the condition (N) is equivalent with the Sard condition (i.e. \(Jf(x)=0\) a.e. on an open set \(A\subseteq G\), yields \(\text{meas } f(A) =0\)).
For a continuous \(f\in W^{1,2} (G,\mathbb{R}^ 2)\), the condition (N) is equivalent with \(f\) almost open in \(G\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26B10 Implicit function theorems, Jacobians, transformations with several variables
55M25 Degree, winding number
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